# This is insane

Why does the media insist on reporting exchange rates this way?

Off topic, but if anyone here is good at statistics, I have a question.  I was regressing monthly stock prices on gold stocks during the late 1920s and early 1930s.  I’d like to show that the variables only became correlated during periods of exchange rate crisis (after mid-1931.)

[Update:  I actually regressed first differences of stock prices and gold stocks.]

I’ll provide the output and you tell me if the difference is significant:

Before mid-1931:  Adj R2 = .000,  t-stat =  0.55,  D-W = 1.26   n= 53

After mid-1931:    Adj R2 = .201     t-stat = 2.51,  D-W =1.81   n=22

Everything looks good to me except the Durbin-Watson from the first regression. Here’s my question.  On the one hand you could argue that the results are not meaningful, as the first D-W is too low.  On the other you could argue that the second regression is fine, and the low D-W on the first doesn’t matter because the R2 is zero anyway, so there’s no correlation at all, nothing to “correct” (which is what I’d like to believe.)

I ask because a reviewer says I can’t use the regressions, and I don’t want to do it all over (I don’t even know where the data is.)  Can I make any kind of claim here?

Will Christmas come early?

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28 Responses to “This is insane”

1. Todd
24. December 2013 at 12:04

Regarding the currencies, they report them based on market conventions. In FX markets, Euro, Pound, Aussie, and the New Zealand dollar are all quoted in amount of USD per the base current currency. Not sure why that convention came about, though…

2. Charlie
24. December 2013 at 12:41

You have stock prices as opposed to stock returns? If both X and Y are unit roots (drifting up over time), your t-stat isn’t meaningful. They’ll be spuriously correlated due to the time trend.

3. ssumner
24. December 2013 at 13:00

Charlie, I used monthly first differences, so there’s not really a time trend problem because the monthly fluctuations over such a short period (2 to 4 years) dominates any trend.

4. The Cable Guy
24. December 2013 at 13:33

Todd is correct that these are the market convention, not an arbitrary choice by the news outlets. As for why the convention grew up that way, the only explanation I’ve ever heard that makes any sense is that the general goal was to maximize the chance that quotes were > 1. Currently the only quotes against USD where USD is not the base are AUD, EUR, GBP, NZD.

5. Ironman
24. December 2013 at 13:37

Your “other hand” argument is the stronger of the two. The main trouble I see is that with an R2 = .201 for the post-1931 data, you’re still dealing with a weak correlation, which may be what is giving your reviewer fits.

The question that needs to be answered is “is it significant?” I think the answer in this case would be yes, because you’re dealing with what is in effect a change in state. It’s not suitable for making predictions (since clearly, there are other confounding factors at work), but it would perhaps be suitable for establishing that a difference with the preceding period exists, which is really what you’re after in this case.

6. Geoff
24. December 2013 at 13:54

The post-1931 result is statistically significant, because for (n-1) = 21 degrees of freedom, and 5% level of significance, the critical t-value is 2.074, which is less than your sample t-value of 2.51.

The R2 is 0.201, which in finance is above the typically minimum acceptable coeff. of determination.

7. ssumner
24. December 2013 at 13:56

Thanks Todd.

Thanks Ironman, An R2 of .20 can be considered economically significant in a regression of first differences, if you are simply trying to show that X affects Y, not that X is the only variable affecting Y.

Glad to hear my “on the other hand” was OK. That’s what I’m hoping to claim.

8. ssumner
24. December 2013 at 13:57

Thanks Geoff.

9. ssumner
24. December 2013 at 14:02

Whatever the reason, Yahoo should report all the exchange rates as the foreign currency price of dollars. That’s the only way to make the changes comprehensible. And the changes are what we care about—did the dollar rise or fall that day.

24. December 2013 at 14:03

“I ask because a reviewer says I can’t use the regressions, and I don’t want to do it all over (I don’t even know where the data is.) Can I make any kind of claim here?”

Can’t you just report the results you have here? Even if there’s a problem with residual autocorrelation the fact that the first correlation isn’t significant is highly suggestive. It’s very unlikely to change if you manage to address the problem. (The R-squared is zero for heaven’s sake!)

On the other hand, what’s stopping you from just finding the data and redoing the damned regression? Get to work!

11. Rick G
24. December 2013 at 14:11

You could make a stronger claim if you did a change-point analysis, assessing at what month and year the correlation went from zero to non-zero. This involves bootstrap simulation with the data but if you want to just post the time-series or send them to me I can run it.

12. Kevin Erdmann (aka kebko)
24. December 2013 at 14:11

I believe that the convention on currency quotes came about because up until the 1970’s, the British pound was divisible by fractions instead of decimals, so that it was difficult to quote currencies in terms of the pound. So, to this day, the convention stands that the currencies related to Great Britain are stated this way, even though the fractional issue doesn’t exist. It’s a QWERTY thing. It’s not the media. The media does it because the banks do it.

Fx quotes are so funny. Our brains our anchored to deal with prices in terms of how many dollars per widget. So, in Fx, where you have to keep switching between dollars per “widget” and “widgets” per dollar, the typical brain (including mine) turns to mush. I always get it wrong if I try to use intuition. The different quoting methods just makes the whole thing even worse.

13. Joseph
24. December 2013 at 14:31

Even if there is some kind of autocorrelation as evidenced by the D-W that would only serve to inflate the t-stat but even this inflated t-stat is insignificant and as noted the r^2 is still 0.

14. ssumner
24. December 2013 at 14:33

Thanks Mark.

Thanks Rick, I’ll see if I can find the data.

Kevin, Yes, but Yahoo would do us all a big favor if they’d start reporting the data in one way only. Whenever I look at the forex charts I have no idea if the dollar went up or down, and I’m too lazy to figure it out. If they’d report all the data the same way we could just glance at the chart and know what happened.

15. Lorenzo from Oz
24. December 2013 at 16:07

This is the age of the tailored internet. Shouldn’t it just work that your country’s currency is the base depending on where you read it from?

16. Robert Sams
25. December 2013 at 02:52

On the forex, why not use DXY, the trade-weighed dollar index which has a futures contact on ICE. Free daily ts found here: http://www.quandl.com/futures/us-dollar-index-futures

The quotes on the majors are convention and worth committing to memory. Only EUR, GBP, and AUD quote in their terms, so if you just think of global pretensions of the ccy’s underlying politics.. You’ll never forget the convention.

On the regression thing, I wouldn’t be too bothered about the test statistics or the regression itself for that matter. I’d be more interested in a more granular identification assumption (binning the ts into crisis/no crisis regimes). If there is anything to the relationship it will jump out in a simple comparison of regime scatter plots. Those who go on about statistical significance when dealing with financial ts have lost sight of the subject matter.. Need robust methods here.

Merry Christmas!

17. ssumner
25. December 2013 at 05:47

Lorenzo, Good point.

Thanks Robert.

18. Matthias
25. December 2013 at 07:44

First thing I learned in time series analysis is, that DW is a very poor indicator of autocorrelation. If you need it to publish, ok, but might be worth to look at other indicators.

19. Matthias
25. December 2013 at 07:46

…such as the Breusch-Godfrey test.

20. dtoh
26. December 2013 at 01:54

The market convention has always been that way. You would totally confuse traders and anybody else who deals regularly in fx if you did it any other way.

Historically everything used to be quoted with GBP as the base currency so the USD/GBP continues to be quoted that way. Pretty much everything else is quoted with USD as the base currency.

The main exception is the Euro. I don’t know why this is. I suspect it has something to do with European pretentiousness.

21. Doug M
26. December 2013 at 10:02

FX conventions…

For the first 20 years of my life we used to say that the Canadian dollar was worth 70 cents. Sometime in the 90s, it flip-flopped, and we said that the USD was worth 1.35 CAD.

It used to be that English speaking countries were quoted in:
Unit of currency / USD (GBP, AUD, NZD, CAD) and non English speaking coutries were in USD / unit FX .

When the Euro was introduced the ECB announced its preference to be quote itself in EUR / USD.

On regressions.. you cannot regress price levels, only price changes.
Regression 1, is no observed correlation. You cannot say that the correlation doesn’t exist, just that you were unable to see it.
Regression 2, is still pretty weakly correlated. I would not make too much out of a non-correlation turning into a weakly correlated series.

22. Cthorm
26. December 2013 at 10:19

Scott,

You’re right, Yahoo would be smart to offer more ways to view exchange rates than “market convention”. Market convention even annoys practitioners. Pricing everything in one base currency isn’t the best way though. Bloomberg has this down pat…

They use a a table. The leftmost row is the denominator, the top row is the numerator. You can choose between displaying ‘last price’ or ‘% change’, and these are all highlighted using a red-green heatmap. This way lets you easily see asymmetries in the movements between different FX cross rates. For example, today the SEK has strengthened against all the major currencies, but it has strengthened most against NZD, AUD, CAD, and JPY.

Re: your paper dilemna, Ironman has it right. You’re showing a change state, so a movement from one variable having a zero explanatory power to 21% is interesting.

23. Ironman
27. December 2013 at 04:50

Data source you might be interested in: xe.com. Historical data only goes back to 1995, and they do charts with up to 10 years of data for individual currency conversions as well.

24. Alex
27. December 2013 at 04:58

On the statistics thing:
I feel like the question you are asking is: “Can the t-statistics/R^2 be deflated by serial correlation in the residual?” If so, then the change in your results could be caused by some shift in the error structure and not by a change in the underlying relationship between the two variables. Unfortunately the answer to the question is “Yes”. Serial correlation is in essence a kind of misspecification problem (Greene 253, 5th ed), and its effect is likely lowering the amount of variation that can be explained by the dependent variable (the in model variation).

I’m not entirely sure that this matters: Your stated null is that the correlation before and after 1931 are the same. Thus to my (admittedly simplistic) mind the obvious thing to do is compute the correlation coefficient in each sample and test whether or not they are different.
http://vassarstats.net/rdiff.html
The D-W matters allot more if you are asking causal questions, but you don’t seem to be…

These correlation coefficients can be extracted from the statistics you posted with a bit of least squares algebra (presuming this is from a simple regression!), and I got them as .076 and .489 respectively, which viewed as a one-tailed test via the website above is significant at the 5% level.

(I don’t vouch for the accuracy of my calculations, here’s how I got them though: http://arhouser.net/odds-ends/recovering_r.pdf)

25. Mike Freimuth
27. December 2013 at 23:39

I feel the same way about exchange rates. I always just look at the WSJ dollar index. Other similar indexes exist and I think are the simplest solution to your problem. For the record, the dollar got weaker but yes it always takes me a minute to work it out and sometimes it isn’t clear since obviously sometimes they move relative to each other.

In addition to doing some one way and others another, I’ve always thought they should be labeled the opposite way (USD/JPY should mean the number of dollars you can buy with one yen). I convinced myself that what they do makes sense but if I hadn’t known which one was more valuable to begin with, I don’t know if I ever would have made sense of it.

26. Sshankar
28. December 2013 at 03:26

As for your stats question, if the correlation among erros is positive (which should be easy to see given the number of points). If postively correlated the t-stat for both regression is overstated. So there definitely was no correlation betwwen the vaiables. Also as it doesn’t bias the parameter estimate, so it is likely that the coefficients you got are correct. Also given the nature of your problem you coudl use the Chow test, to show that there is a structural break (the regression changes). You can also use a slope dummy on gold-stock. Ofcourse the second method would require running the entire regression.
On the other hand just from the estimates, the 2nd regression has a siginificant t-stat, while the first is too weak. I see no reason for these results not presenting a defendable case for your hypothesis.

27. ssumner
28. December 2013 at 11:31

Everyone, Thanks for the stats info. Very helpful.

28. Alek
29. December 2013 at 09:22

It could be that you have over differenced your data set and that is why you are seeing these autocorrelations.

See for instance “a brief parable of over-Differencing” by cochrane